Eigenvalues and Markov Chains


 Peregrine Noah Shelton
 1 years ago
 Views:
Transcription
1 Eigenvalues and Markov Chains Will Perkins April 15, 2013
2 The Metropolis Algorithm Say we want to sample from a different distribution, not necessarily uniform. Can we change the transition rates in such a way that our desired distribution is stationary? Amazingly, yes. Say we have a distribution π over X so that π(x) = w(x) y X w(y) I.e. we know the proportions but not the normalizing constant (and X is much too big to compute it).
3 The Metropolis Algorithm MetropolisHastings Algorithm 1 Create a graph structure on X so the graph is connected and has maximum degree D. 2 Define the following transition probabilities: 1 p(x, y) = 1 2D (max{w(y)/w(x), 1}) if x and y are neighbors. 2 p(x, y) = 0 if x and y are not neighbors 3 p(x, x) = 1 y x p(x, y) 3 Check that this Markov chain is irreducible, aperiodic, reversible and has stationary distribution π.
4 Example Say we want to sample large independent sets from a graph G. I.e. P(I ) = λ I where Z = J λ J where the sum is over all independent sets. Note that this distribution gives more weight to the largest independent sets. Use the Metropolis Algorithm to find a Markov Chain with this distribution as the stationary distribution. Z
5 Linear Algebra Recall some facts from linear algebra: If A is a real symmetric, n n matrix, then A has real eigenvalues and there exists an orthonormal basis of R n consisting of eigenvectors of A. The eigenvalues of A n are the eigenvalues of A raised to the n Rayleigh Quotient form of eigenvalues
6 PerronFrobenius Theorem Theorem Let A > 0 be a matrix with all positive entries. Then there exists an eigenvalue λ 0 > 0 with eigenvector x 0 all of whose entries are positive so that 1 If λ λ 0 is another eigenvalue of A then λ < λ 0. 2 λ 0 has algebraic and geometric multiplicity 1
7 PerronFrobenius Theorem Proof: Define a set of real numbers Λ = {λ : Ax λx for some x 0}. Show that Λ [0, M] for some M. Then let λ 0 = max Λ. From the definition of Λ, there exists an x 0 0 so that Ax 0 λ 0 x 0. Suppose Ax 0 λx 0. Then let y = Ax 0 and A(y λ 0 x 0 ) = Ay λ 0 y > 0 since A > 0. But this is a contradiction. So Ax 0 = λ 0 x 0.
8 PerronFrobenius Theorem Now pick an eigenvalue λ λ 0 with eigenvector x. Then and so λ λ 0. A x Ax = λx = λ x Finally, we show that there is no other eigenvalue λ = λ 0. Consider A δ = A = δi for small enough δ so the matrix is still positive. A δ has eigenvalues λ 0 δ and λ δ, and λ 0 δ λ δ. But if λ λ 0 is on the same circle in the complex plane as λ 0, this is a contradiction. [picture]
9 PerronFrobenius Theorem Finally, we address the multiplicity. Say x and y are linearly independent eigenvectors with eigenvalue λ 0. Then find α so that x + α y has nonnegative entries, but at least one 0 entry. But since A > 0 and A(x + αy) = λ(x + αy) there is a contradiction.
10 Application to Markov Chains Check: the conclusions of the PerrronFrobenius theorem hold for the transition matrix of a finite, aperiodic, irreducible Markov chain.
11 Rate of Convergence Theorem Consider the transition matrix P of a symmetric, aperiodic, irreducible Markov Chain on n states. Let µ be the uniform (stationary) distribution. Let λ 1 = 1 be the largest eigenvalue and λ 2 the secondlargest in absolute values. Then π (x) m µ TV n λ 2 m Proof: Start with the Jordan Canonical form of the matrix P. (A generalization of diagonalizing  we ll assume P is diagonalizable), i.e. D = UPU 1 The rows of U are the left eigenvectors of P and the columns of U 1 are the right eigenvectors.
12 Rate of Convergence Order the eigenvalues 1 = λ 1 > λ 2 >.... The left eigenvector of λ 1 is the stationary distribution vector. The first right eigenvector is the all 1 s vector. Now write P n = U 1 D n U. Write π 0 is the eigenvector basis: π 0 = µ + c 2 u c n u n and π m = π 0 P m = µ + where λ j λ 2 < 1. n c j λ m j u j j=2
13 Eigenvalues of Graphs The adjacency matrix A of a graph G is the matrix whose i, jth entry is 1 if (i, j) E(G). The normalized adjacency matrix turns this into a stochastic matrix  for example, if G is dregular, we divide A by d. For dregular graph, with normalized adjancey matrix A, What is λ 1? What does A correspond to in terms of Markov Chains? What does it mean if λ 2 = 1? What does it mean if λ n = 1?
14 Cheeger s Inequality For a dregular graph, define the edge expansion of a cut S V as: h(s) = E(S, S c ) d min{ S, S c } The edge expansion of a graph G is h(g) = min S V h(s)
15 Cheeger s Inequality Theorem (Cheeger s Inequality) Let 1 = λ 1 λ 2... be the eigenvalues of the random walk on the dregular graph G. Then 1 λ 2 2 h(g) 2(1 λ 2 ) What does this say about mixing times of random walks on graphs?
16 Ehrenfest Urn What are the eigenvalues and eigenvectors of the Ehrenfest Urn?
1 Eigenvalues and Eigenvectors
Math 20 Chapter 5 Eigenvalues and Eigenvectors Eigenvalues and Eigenvectors. Definition: A scalar λ is called an eigenvalue of the n n matrix A is there is a nontrivial solution x of Ax = λx. Such an x
More informationChapter 6. Orthogonality
6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More information(67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7
(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition
More informationNotes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
More informationSimilar matrices and Jordan form
Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive
More informationSolutions to Linear Algebra Practice Problems
Solutions to Linear Algebra Practice Problems. Find all solutions to the following systems of linear equations. (a) x x + x 5 x x x + x + x 5 (b) x + x + x x + x + x x + x + 8x Answer: (a) We create the
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationPerron Frobenius theory and some extensions
and some extensions Department of Mathematics University of Ioannina GREECE Como, Italy, May 2008 History Theorem (1) The dominant eigenvalue of a matrix with positive entries is positive and the corresponding
More informationIntroduction to Markov Chain Monte Carlo
Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem
More information1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each)
Math 33 AH : Solution to the Final Exam Honors Linear Algebra and Applications 1. True/False: Circle the correct answer. No justifications are needed in this exercise. (1 point each) (1) If A is an invertible
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationPresentation 3: Eigenvalues and Eigenvectors of a Matrix
Colleen Kirksey, Beth Van Schoyck, Dennis Bowers MATH 280: Problem Solving November 18, 2011 Presentation 3: Eigenvalues and Eigenvectors of a Matrix Order of Presentation: 1. Definitions of Eigenvalues
More informationMATH 551  APPLIED MATRIX THEORY
MATH 55  APPLIED MATRIX THEORY FINAL TEST: SAMPLE with SOLUTIONS (25 points NAME: PROBLEM (3 points A web of 5 pages is described by a directed graph whose matrix is given by A Do the following ( points
More informationMatrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo
Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for
More informationNotes on Jordan Canonical Form
Notes on Jordan Canonical Form Eric Klavins University of Washington 8 Jordan blocks and Jordan form A Jordan Block of size m and value λ is a matrix J m (λ) having the value λ repeated along the main
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationLecture12 The PerronFrobenius theorem.
Lecture12 The PerronFrobenius theorem. 1 Statement of the theorem. 2 Proof of the Perron Frobenius theorem. 3 Graphology. 3 Asymptotic behavior. The nonprimitive case. 4 The Leslie model of population
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationOctober 3rd, 2012. Linear Algebra & Properties of the Covariance Matrix
Linear Algebra & Properties of the Covariance Matrix October 3rd, 2012 Estimation of r and C Let rn 1, rn, t..., rn T be the historical return rates on the n th asset. rn 1 rṇ 2 r n =. r T n n = 1, 2,...,
More informationDefinition: A square matrix A is block diagonal if A has the form A 1 O O O A 2 O A =
The question we want to answer now is the following: If A is not similar to a diagonal matrix, then what is the simplest matrix that A is similar to? Before we can provide the answer, we will have to introduce
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the yaxis We observe that
More informationEigenvalues and eigenvectors of a matrix
Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a nonzero column vector V such that AV = λv then λ is called an eigenvalue of A and V is
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More information[1] Diagonal factorization
8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:
More informationCheeger Inequalities for General EdgeWeighted Directed Graphs
Cheeger Inequalities for General EdgeWeighted Directed Graphs TH. Hubert Chan, Zhihao Gavin Tang, and Chenzi Zhang The University of Hong Kong {hubert,zhtang,czzhang}@cs.hku.hk Abstract. We consider
More informationNumerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course G63.2010.001 / G22.2420001, Fall 2010 September 30th, 2010 A. Donev (Courant Institute)
More informationLecture 3: Linear Programming Relaxations and Rounding
Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationLinear Algebra Review. Vectors
Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka kosecka@cs.gmu.edu http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa Cogsci 8F Linear Algebra review UCSD Vectors The length
More informationInner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationInner products on R n, and more
Inner products on R n, and more Peyam Ryan Tabrizian Friday, April 12th, 2013 1 Introduction You might be wondering: Are there inner products on R n that are not the usual dot product x y = x 1 y 1 + +
More informationIEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS
IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)
More information(January 14, 2009) End k (V ) End k (V/W )
(January 14, 29) [16.1] Let p be the smallest prime dividing the order of a finite group G. Show that a subgroup H of G of index p is necessarily normal. Let G act on cosets gh of H by left multiplication.
More informationLinear Algebra Notes
Linear Algebra Notes Chapter 19 KERNEL AND IMAGE OF A MATRIX Take an n m matrix a 11 a 12 a 1m a 21 a 22 a 2m a n1 a n2 a nm and think of it as a function A : R m R n The kernel of A is defined as Note
More informationThe Second Eigenvalue of the Google Matrix
0 2 The Second Eigenvalue of the Google Matrix Taher H Haveliwala and Sepandar D Kamvar Stanford University taherh,sdkamvar @csstanfordedu Abstract We determine analytically the modulus of the second eigenvalue
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More information. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2
4. Basic feasible solutions and vertices of polyhedra Due to the fundamental theorem of Linear Programming, to solve any LP it suffices to consider the vertices (finitely many) of the polyhedron P of the
More informationBig Data Technology Motivating NoSQL Databases: Computing Page Importance Metrics at Crawl Time
Big Data Technology Motivating NoSQL Databases: Computing Page Importance Metrics at Crawl Time Edward Bortnikov & Ronny Lempel Yahoo! Labs, Haifa Class Outline Linkbased page importance measures Why
More informationLecture 1: Schur s Unitary Triangularization Theorem
Lecture 1: Schur s Unitary Triangularization Theorem This lecture introduces the notion of unitary equivalence and presents Schur s theorem and some of its consequences It roughly corresponds to Sections
More information5. Orthogonal matrices
L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 51 Orthonormal
More informationTHE $25,000,000,000 EIGENVECTOR THE LINEAR ALGEBRA BEHIND GOOGLE
THE $5,,, EIGENVECTOR THE LINEAR ALGEBRA BEHIND GOOGLE KURT BRYAN AND TANYA LEISE Abstract. Google s success derives in large part from its PageRank algorithm, which ranks the importance of webpages according
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationQuadratic Functions, Optimization, and Quadratic Forms
Quadratic Functions, Optimization, and Quadratic Forms Robert M. Freund February, 2004 2004 Massachusetts Institute of echnology. 1 2 1 Quadratic Optimization A quadratic optimization problem is an optimization
More informationDiagonalisation. Chapter 3. Introduction. Eigenvalues and eigenvectors. Reading. Definitions
Chapter 3 Diagonalisation Eigenvalues and eigenvectors, diagonalisation of a matrix, orthogonal diagonalisation fo symmetric matrices Reading As in the previous chapter, there is no specific essential
More informationRandom access protocols for channel access. Markov chains and their stability. Laurent Massoulié.
Random access protocols for channel access Markov chains and their stability laurent.massoulie@inria.fr Aloha: the first random access protocol for channel access [Abramson, Hawaii 70] Goal: allow machines
More informationLECTURE 4. Last time: Lecture outline
LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationSearch engines: ranking algorithms
Search engines: ranking algorithms Gianna M. Del Corso Dipartimento di Informatica, Università di Pisa, Italy ESP, 25 Marzo 2015 1 Statistics 2 Search Engines Ranking Algorithms HITS Web Analytics Estimated
More informationConductance, the Normalized Laplacian, and Cheeger s Inequality
Spectral Graph Theory Lecture 6 Conductance, the Normalized Laplacian, and Cheeger s Inequality Daniel A. Spielman September 21, 2015 Disclaimer These notes are not necessarily an accurate representation
More information1 2 3 1 1 2 x = + x 2 + x 4 1 0 1
(d) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b. 1 2 3 1 1 2 x = + x 2 + x 4 1 0 0 1 0 1 2. (11 points) This problem finds the curve y = C + D 2 t which
More informationThe Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations
The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential
More informationNotes on Symmetric Matrices
CPSC 536N: Randomized Algorithms 201112 Term 2 Notes on Symmetric Matrices Prof. Nick Harvey University of British Columbia 1 Symmetric Matrices We review some basic results concerning symmetric matrices.
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationThe Hadamard Product
The Hadamard Product Elizabeth Million April 12, 2007 1 Introduction and Basic Results As inexperienced mathematicians we may have once thought that the natural definition for matrix multiplication would
More informationContinuity of the Perron Root
Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North
More informationCSE 494 CSE/CBS 598 (Fall 2007): Numerical Linear Algebra for Data Exploration Clustering Instructor: Jieping Ye
CSE 494 CSE/CBS 598 Fall 2007: Numerical Linear Algebra for Data Exploration Clustering Instructor: Jieping Ye 1 Introduction One important method for data compression and classification is to organize
More information22 Matrix exponent. Equal eigenvalues
22 Matrix exponent. Equal eigenvalues 22. Matrix exponent Consider a first order differential equation of the form y = ay, a R, with the initial condition y) = y. Of course, we know that the solution to
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 6. Eigenvalues and Singular Values In this section, we collect together the basic facts about eigenvalues and eigenvectors. From a geometrical viewpoint,
More informationWe know a formula for and some properties of the determinant. Now we see how the determinant can be used.
Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we
More informationLinear Dependence Tests
Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks
More informationThe Geometry of Graphs
The Geometry of Graphs Paul Horn Department of Mathematics University of Denver May 21, 2016 Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. Graphs Ultimately, I want
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationApplied Linear Algebra I Review page 1
Applied Linear Algebra Review 1 I. Determinants A. Definition of a determinant 1. Using sum a. Permutations i. Sign of a permutation ii. Cycle 2. Uniqueness of the determinant function in terms of properties
More information7 Communication Classes
this version: 26 February 2009 7 Communication Classes Perhaps surprisingly, we can learn much about the longrun behavior of a Markov chain merely from the zero pattern of its transition matrix. In the
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationLinear Algebra Problems
Math 504 505 Linear Algebra Problems Jerry L. Kazdan Note: New problems are often added to this collection so the problem numbers change. If you want to refer others to these problems by number, it is
More informationBonusmalus systems and Markov chains
Bonusmalus systems and Markov chains Dutch car insurance bonusmalus system class % increase new class after # claims 0 1 2 >3 14 30 14 9 5 1 13 32.5 14 8 4 1 12 35 13 8 4 1 11 37.5 12 7 3 1 10 40 11
More informationRanking on Data Manifolds
Ranking on Data Manifolds Dengyong Zhou, Jason Weston, Arthur Gretton, Olivier Bousquet, and Bernhard Schölkopf Max Planck Institute for Biological Cybernetics, 72076 Tuebingen, Germany {firstname.secondname
More informationBASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM
BASIC THEORY AND APPLICATIONS OF THE JORDAN CANONICAL FORM JORDAN BELL Abstract. This paper gives a basic introduction to the Jordan canonical form and its applications. It looks at the Jordan canonical
More informationEssentials of Stochastic Processes
i Essentials of Stochastic Processes Rick Durrett 70 60 50 10 Sep 10 Jun 10 May at expiry 40 30 20 10 0 500 520 540 560 580 600 620 640 660 680 700 Almost Final Version of the 2nd Edition, December, 2011
More informationVector Spaces II: Finite Dimensional Linear Algebra 1
John Nachbar September 2, 2014 Vector Spaces II: Finite Dimensional Linear Algebra 1 1 Definitions and Basic Theorems. For basic properties and notation for R N, see the notes Vector Spaces I. Definition
More informationsome algebra prelim solutions
some algebra prelim solutions David Morawski August 19, 2012 Problem (Spring 2008, #5). Show that f(x) = x p x + a is irreducible over F p whenever a F p is not zero. Proof. First, note that f(x) has no
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More information1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let
Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as
More informationMath 4310 Handout  Quotient Vector Spaces
Math 4310 Handout  Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable
More informationLecture 5 Principal Minors and the Hessian
Lecture 5 Principal Minors and the Hessian Eivind Eriksen BI Norwegian School of Management Department of Economics October 01, 2010 Eivind Eriksen (BI Dept of Economics) Lecture 5 Principal Minors and
More informationSPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BIDIAGONAL REPRESENTATIONS FOR PHASE TYPE DISTRIBUTIONS AND MATRIXEXPONENTIAL DISTRIBUTIONS
Stochastic Models, 22:289 317, 2006 Copyright Taylor & Francis Group, LLC ISSN: 15326349 print/15324214 online DOI: 10.1080/15326340600649045 SPECTRAL POLYNOMIAL ALGORITHMS FOR COMPUTING BIDIAGONAL
More informationActually Doing It! 6. Prove that the regular unit cube (say 1cm=unit) of sufficiently high dimension can fit inside it the whole city of New York.
1: 1. Compute a random 4dimensional polytope P as the convex hull of 10 random points using rand sphere(4,10). Run VISUAL to see a Schlegel diagram. How many 3dimensional polytopes do you see? How many
More informationSF2940: Probability theory Lecture 8: Multivariate Normal Distribution
SF2940: Probability theory Lecture 8: Multivariate Normal Distribution Timo Koski 24.09.2015 Timo Koski Matematisk statistik 24.09.2015 1 / 1 Learning outcomes Random vectors, mean vector, covariance matrix,
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationTHE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE. Alexander Barvinok
THE NUMBER OF GRAPHS AND A RANDOM GRAPH WITH A GIVEN DEGREE SEQUENCE Alexer Barvinok Papers are available at http://www.math.lsa.umich.edu/ barvinok/papers.html This is a joint work with J.A. Hartigan
More informationUNCOUPLING THE PERRON EIGENVECTOR PROBLEM
UNCOUPLING THE PERRON EIGENVECTOR PROBLEM Carl D Meyer INTRODUCTION Foranonnegative irreducible matrix m m with spectral radius ρ,afundamental problem concerns the determination of the unique normalized
More informationElementary Linear Algebra
Elementary Linear Algebra Kuttler January, Saylor URL: http://wwwsaylororg/courses/ma/ Saylor URL: http://wwwsaylororg/courses/ma/ Contents Some Prerequisite Topics Sets And Set Notation Functions Graphs
More information4. MATRICES Matrices
4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:
More informationThe PerronFrobenius theorem.
Chapter 9 The PerronFrobenius theorem. The theorem we will discuss in this chapter (to be stated below) about matrices with nonnegative entries, was proved, for matrices with strictly positive entries,
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationDefinition of a Linear Program
Definition of a Linear Program Definition: A function f(x 1, x,..., x n ) of x 1, x,..., x n is a linear function if and only if for some set of constants c 1, c,..., c n, f(x 1, x,..., x n ) = c 1 x 1
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationAu = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
More information3. Evaluate the objective function at each vertex. Put the vertices into a table: Vertex P=3x+2y (0, 0) 0 min (0, 5) 10 (15, 0) 45 (12, 2) 40 Max
SOLUTION OF LINEAR PROGRAMMING PROBLEMS THEOREM 1 If a linear programming problem has a solution, then it must occur at a vertex, or corner point, of the feasible set, S, associated with the problem. Furthermore,
More informationLINEAR ALGEBRA AND MATRICES M3P9
LINEAR ALGEBRA AND MATRICES M3P9 December 11, 2000 PART ONE. LINEAR TRANSFORMATIONS Let k be a field. Usually, k will be the field of complex numbers C, but it can also be any other field, e.g., the field
More informationPart 1: Link Analysis & Page Rank
Chapter 8: Graph Data Part 1: Link Analysis & Page Rank Based on Leskovec, Rajaraman, Ullman 214: Mining of Massive Datasets 1 Exam on the 5th of February, 216, 14. to 16. If you wish to attend, please
More informationSolutions to Math 51 First Exam January 29, 2015
Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not
More informationLecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs
CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More information